This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, the proposed method incorporates backward-step information to significantly improve both accuracy and stability. The derivation employs multistep collocation and interpolation techniques, with the Chebyshev polynomial of the first kind serving as the basis function. This choice of basis function is particularly effective for approximating oscillatory behavior. Unknown parameters are efficiently determined using Gaussian elimination. The resulting continuous scheme is then evaluated at selected points to obtain the discrete block method. A detailed theoretical analysis establishes the method's order, error constant, consistency, and zero-stability, collectively confirming its convergence. To validate its performance, the method is applied to standard oscillatory test problems. Numerical results demonstrate that the proposed method achieves superior accuracy compared to existing methods in the literature. The findings confirm that this backward-step hybrid block method offers a robust, reliable, and computationally efficient solution for oscillatory IVPs. Its accuracy and stability make it a valuable tool for researchers and practitioners dealing with oscillatory problems in engineering and the physical sciences.
| Published in | American Journal of Applied Mathematics (Volume 14, Issue 5) |
| DOI | 10.11648/j.ajam.20261405.18 |
| Page(s) | 348-358 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Linear Multi-step Method, Hybrid Points, Chebyshev Polynomial, Oscillatory, Zero-Stability
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APA Style
Titilayo, O. B., Afeez, A., Lukuman, M. A., Moses, O. (2026). Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems. American Journal of Applied Mathematics, 14(5), 348-358. https://doi.org/10.11648/j.ajam.20261405.18
ACS Style
Titilayo, O. B.; Afeez, A.; Lukuman, M. A.; Moses, O. Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems. Am. J. Appl. Math. 2026, 14(5), 348-358. doi: 10.11648/j.ajam.20261405.18
AMA Style
Titilayo OB, Afeez A, Lukuman MA, Moses O. Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems. Am J Appl Math. 2026;14(5):348-358. doi: 10.11648/j.ajam.20261405.18
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Download@article{10.11648/j.ajam.20261405.18,
author = {Olabode Bola Titilayo and Abidemi Afeez and Momoh Adelegan Lukuman and Oluwadamilola Moses},
title = {Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems},
journal = {American Journal of Applied Mathematics},
volume = {14},
number = {5},
pages = {348-358},
doi = {10.11648/j.ajam.20261405.18},
url = {https://doi.org/10.11648/j.ajam.20261405.18},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.18},
abstract = {This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, the proposed method incorporates backward-step information to significantly improve both accuracy and stability. The derivation employs multistep collocation and interpolation techniques, with the Chebyshev polynomial of the first kind serving as the basis function. This choice of basis function is particularly effective for approximating oscillatory behavior. Unknown parameters are efficiently determined using Gaussian elimination. The resulting continuous scheme is then evaluated at selected points to obtain the discrete block method. A detailed theoretical analysis establishes the method's order, error constant, consistency, and zero-stability, collectively confirming its convergence. To validate its performance, the method is applied to standard oscillatory test problems. Numerical results demonstrate that the proposed method achieves superior accuracy compared to existing methods in the literature. The findings confirm that this backward-step hybrid block method offers a robust, reliable, and computationally efficient solution for oscillatory IVPs. Its accuracy and stability make it a valuable tool for researchers and practitioners dealing with oscillatory problems in engineering and the physical sciences.},
year = {2026}
}
TY - JOUR T1 - Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems AU - Olabode Bola Titilayo AU - Abidemi Afeez AU - Momoh Adelegan Lukuman AU - Oluwadamilola Moses Y1 - 2026/09/28 PY - 2026 N1 - https://doi.org/10.11648/j.ajam.20261405.18 DO - 10.11648/j.ajam.20261405.18 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 348 EP - 358 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20261405.18 AB - This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, the proposed method incorporates backward-step information to significantly improve both accuracy and stability. The derivation employs multistep collocation and interpolation techniques, with the Chebyshev polynomial of the first kind serving as the basis function. This choice of basis function is particularly effective for approximating oscillatory behavior. Unknown parameters are efficiently determined using Gaussian elimination. The resulting continuous scheme is then evaluated at selected points to obtain the discrete block method. A detailed theoretical analysis establishes the method's order, error constant, consistency, and zero-stability, collectively confirming its convergence. To validate its performance, the method is applied to standard oscillatory test problems. Numerical results demonstrate that the proposed method achieves superior accuracy compared to existing methods in the literature. The findings confirm that this backward-step hybrid block method offers a robust, reliable, and computationally efficient solution for oscillatory IVPs. Its accuracy and stability make it a valuable tool for researchers and practitioners dealing with oscillatory problems in engineering and the physical sciences. VL - 14 IS - 5 ER -
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
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